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You count the number of sets, and the number of operations for each set. You can also think about action by scalar or basis vectors.

So monoids groups and rings have one set. Groups and monoids have one operation, while rings have two operations.

Modules, vector spaces, and algebras have two sets. I call them the scalars and the vectors, for today. The scalars form a ring or field; the vectors form an abelian group, or even a ring themselves.

Monoid rings are interesting, they are hard to fit into my previous scheme, but are very important. They are an algebra, in the sense that they have two sets, a ring of scalars, and a ring of vectors. But further, their defining characteristic is that they have a basis of elements which can be multiplied. They include [STRIKE]adjuncts from Galois theory[/STRIKE], polynomial rings et cetera.

So we see algebras are both rings and modules, two different ways to look at them. Every monoid ring is an algebra.

But I wouldn't assume every algebra is a monoid ring, since we may not have a basis?

So this is my new framework as I review algebra texts and think about keeping track of examples.

Also, I sort of see the relevancy of for instance Lang's part 1 is about groups, rings, modules and polynomials, as these are sort of the basic objects as we increase structure, in the sense of number of sets and operations.